Nonlinear statistics is an increasingly active research field at the intersection of geometry, statistics, machine learning and algorithmics. Nonlinear statistics seeks to answer fundamental questions that arise when defining new statistical models and tools for analysis of data that exhibits nonlinearity. Such data is becoming increasingly prevalent in diverse fields including computational anatomy, phylogenetics, and computational biology, and are also receiving increased interest in machine learning, where nonlinearity adds to the possible flexibility of a predictive model. In addition to impact on applied problems, theoretical advances in nonlinear statistics also provides new insight into methodology from traditional linear statistics.

Scientific content:

The course will consist of 5 days with lectures before lunch and exercise sessions after lunch. In addition, students will be expected to read a pre-defined set of scientific articles on nonlinear statistics prior to the course, and write a report on nonlinear statistics and its potential use in their own research field after the course. The course will consist of three modules corresponding to the three external lecturers,namely:

The first module will introduce standard methods in manifold statistics and recent advances on these. Manifolds are the most commonly used nonlinear data spaces, and also the most tractable computationally.

The second module will introduce methods available in nonlinear data spaces that are not manifolds – in other words, they are not smooth. This is relevant for structured data such as trees and graphs, as well as data with degeneracies such as diffusion tensors or non-diffeomorphic registration deformations.

The third module will introduce the elastic shape analysis framework developed by A. Srivastava, which has proven useful in a wide range of applications.

Learning outcome

After participating in this course, the student should:

- Have a strong knowledge of basic concepts in nonlinear statistics (geodesic-based statistics such as Fréchet means and geodesic PCA; diffusion-based statistics)- Be able to implement basic numerical tools for statistics on nonlinear data spaces- Have experience working with several types of data residing in nonlinear data spaces (covariance matrices, trees, shapes)- Be aware of current open problems and developments in nonlinear statistics